By JONATHAN MAHLER and EMILY STEEL

July 16, 2014

On a Saturday in early June, Jeffrey L. Bewkes, the chief executive of Time Warner, got an email from Chase Carey, the president of 21st Century Fox, proposing lunch.

They met a few days later in the executive dining room of the Time-Warner Center. Before dessert was served, Mr. Carey got to the point: His company wanted to buy Time Warner for $80 billion.

Time Warner’s board studied and ultimately rejected the offer, but it nevertheless has thrown the company’s future into question. Mr. Carey’s boss — Rupert Murdoch — is not known for backing off once he has set his sights on his prey; his purchase of Dow Jones also began with a spurned offer. What is more, now that Fox has made its interest in Time Warner known, other suitors may surface.

While the offer from Fox may have been unsolicited and even unwanted, it represented a validation of Mr. Bewkes’s stewardship of the company. Since taking over a struggling Time Warner at the start of 2008, he has carried out an ambitious turnaround strategy, methodically shedding Time Warner’s noncore assets to transform the company from a sprawling media conglomerate into a pure entertainment entity.

Gone are the cable business, the Internet business and the magazine business. Time Warner is now a discrete and lucrative group of television and film properties, including Warner Bros., the movie studio, and HBO, the cable network where Mr. Bewkes began his career. The final spinoff, of the publisher Time Inc., was completed just last month, the day before Mr. Carey and Mr. Bewkes met for lunch.

When Mr. Bewkes became chief executive of Time Warner, the company was still reeling from its disastrous merger with AOL eight years earlier. In his early days, the company’s stock traded as low as $14 a share.

Instead of empire-building and rapid growth, the previous trend in media, his plan from the outset was to enhance the value of the company by shrinking it, according to executives who worked with Mr. Bewkes. The idea was to use the cash from spinning off the various divisions to buy back stock and ultimately drive up Time Warner’s market value.

Internally, some executives agreed with the strategy. Others were wary of dismantling the company that had come to define the age of media behemoths.

Questions loom over whether it was Mr. Bewkes’s intention all along to make the company an acquisition target.

“In this day and age, when you make yourself bite-sized, you’re looking for a sale,” said a former colleague of Mr. Bewkes, who spoke on the condition of anonymity. A spokesman for Time Warner said that setting the company up for a sale was never Mr. Bewkes’s intention.

Jeffrey Bewkes, left, Time Warner chief, and Chase Carey, president of 21st Century Fox, at a cable TV conference in 2012.

Brian Snyder / Reuters

On Wall Street, anyway, the success of Mr. Bewkes’s approach has been unequivocal. Time Warner’s stock price has tripled over the last five years. Today, after the news of the rejected Fox bid, it closed at $83.13 a share, up 17 percent.

Time Warner Cable, which is awaiting regulatory approval for a $45 billion merger with Comcast, was the first division to be spun off under Mr. Bewkes in 2009. A few months later, Time Warner unloaded AOL, putting an unceremonious end to what was already considered one of the most infamous mergers in American business history. And, most recently, Time Warner parted with Time Inc., home to Time, Sports Illustrated and People.

On the notable occasion when Mr. Bewkes tried to expand rather than streamline Time Warner’s business, the results were dismal. In 2008, Time Warner signed off on AOL’s $850 million purchase of the social networking site Bebo. AOL wound up selling the company two years later for a pretax loss of about $1.8 million.

One of Time Warner’s remaining crown jewels is HBO, where Mr. Bewkes — whose total compensation was $32.5 million in fiscal 2013 — began at the company in 1979, earning $20,000 a year. “I tagged Jeff to be the C.E.O. of Time Warner back then,” said one of his early HBO colleagues, Curt Viebranz.

Mr. Bewkes, 62, certainly had the grooming for a high-powered corporate career. He grew up in Darien, Conn., and attended Deerfield Academy, Yale and Stanford Business School. In a creative industry populated by moguls with big personalities, Mr. Bewkes keeps a conspicuously low profile and is known mainly as a “numbers guy.”

“He has an ego,” Mr. Viebranz said, “but I don’t think he lives and breathes being the big media executive.”

Mr. Bewkes became chief executive of HBO in 1995, helping to extend the network’s reach and raise its profile by pushing for more original programming and additional channels. “Sex and the City” and “The Sopranos” became big hits during his tenure. In 2002, he was given additional responsibilities, becoming chairman of the company’s networks and entertainment division. Six years later, Mr. Bewkes succeeded Richard Parsons in the chief executive’s suite.

His current contract with Time Warner runs through 2017 and includes a generous exit package. If the $85-a-share offer were accepted, Mr. Bewkes would stand to make more than $95 million.

News of the Fox bid comes at a time of frenzied consolidation within the media industry, as content providers look to bulk up their must-have TV offerings to gain leverage with increasingly powerful cable providers like Comcast. Time Warner makes an especially attractive takeover target because it owns valuable broadcast rights to professional and college basketball, as well as Major League Baseball.

In a video statement sent to Time Warner employees and posted on YouTube, Mr. Bewkes dismissed Fox’s offer, saying that Time Warner’s board had concluded that his strategic plans would “create significantly more value for the company and our shareholders” than “any proposal that Fox is positioned to offer.” Time Warner said it would elaborate on its plans for growth next month.

People familiar with Time Warner’s management said it was convinced that the company would command a higher price if were potentially put on the market a couple of years later. At that point, too, the field of possible bidders might have widened to include wealthy technology companies like Apple or Google, which are looking at investing more heavily in content. Also, at least two other possible bidders, Comcast and AT&T, cannot enter the fray now because they are already in separate merger negotiations.

Whether Mr. Bewkes and Time Warner’s management will be in a position to decide when the company sells — and to whom — is another matter. Whatever they may think of Fox’s offer, their shareholders may pressure Time Warner’s board to enter talks with Fox and to solicit other bids.

Correction: July 18, 2014

Because of an editing error, an article on Thursday about the revamping of Time Warner that made it an attractive takeover target misspelled, in some copies, the given name of the company’s chief executive. He is Jeffrey L. Bewkes, not Jeffery.